Efficient algorithms for analyzing the singularly perturbed boundary value problems of fractional order
•In this paper, we extended and introduced a new algorithm based on reduced and boundary layer correction method for singularly perturbed boundary value problems of fractional order.•The length of boundary layers are approximated by resemblance functions.•Existence, uniqueness, behavior and stabilit...
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| Vydané v: | Communications in nonlinear science & numerical simulation Ročník 57; s. 136 - 168 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Amsterdam
Elsevier B.V
01.04.2018
Elsevier Science Ltd |
| Predmet: | |
| ISSN: | 1007-5704, 1878-7274 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | •In this paper, we extended and introduced a new algorithm based on reduced and boundary layer correction method for singularly perturbed boundary value problems of fractional order.•The length of boundary layers are approximated by resemblance functions.•Existence, uniqueness, behavior and stability of solution as well as error estimate of the method are investigated.•The obtained results confirm that, the workload of the present method descend to determine the numerical layer length, which means that this method is considerable in comparison with other methods.
In this paper, we were concerned with the description of the singularly perturbed boundary value problems in the scope of fractional calculus. We should mention that, one of the main methods used to solve these problems in classical calculus is the so-called matched asymptotic expansion method. However we shall note that, this was not achievable via the existing classical definitions of fractional derivative, because they do not obey the chain rule which one of the key elements of the matched asymptotic expansion method. In order to accommodate this method to fractional derivative, we employ a relatively new derivative so-called the local fractional derivative. Using the properties of local fractional derivative, we extend the matched asymptotic expansion method to the scope of fractional calculus and introduce a reliable new algorithm to develop approximate solutions of the singularly perturbed boundary value problems of fractional order. In the new method, the original problem is partitioned into inner and outer solution equations. The reduced equation is solved with suitable boundary conditions which provide the terminal boundary conditions for the boundary layer correction. The inner solution problem is next solved as a solvable boundary value problem. The width of the boundary layer is approximated using appropriate resemblance function. Some theoretical results are established and proved. Some illustrating examples are solved and the results are compared with those of matched asymptotic expansion method and homotopy analysis method to demonstrate the accuracy and efficiency of the method. It can be observed that, the proposed method approximates the exact solution very well not only in the boundary layer, but also away from the layer. |
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| Bibliografia: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1007-5704 1878-7274 |
| DOI: | 10.1016/j.cnsns.2017.09.012 |